A batter reaches a century at the Chinnaswamy Stadium and tosses the bat high to a team-mate at the boundary. The bat spins end over end as it flies. The handle whips round, and the blade seems to chase it through the air. Yet the team-mate has no trouble judging where to stand. Somewhere inside that tumbling bat, something is moving in a simple and predictable way.
The model so far
In every chapter so far a body has been a point particle. A block on an incline, a car on a banked road and a satellite in orbit were each a single point with mass. The model worked because we asked only where the body went. A point has no size, so it cannot turn, and it has no shape, so it cannot topple.
This is not a weakness of the laws of motion. It is a choice of what to follow. Newton's second law for a whole body, \(\vec F_{\text{net}} = M\vec a\), gives the acceleration of one special point, the centre of mass. You will meet the centre of mass in detail in the chapter on Centre of Mass. For a uniform rod it is the middle, and for a uniform disc it is the centre.
Watch the two trails in Fig 1.1. The centre C moves on a parabola, exactly as a stone would. Its horizontal velocity stays \(4\,\text{m/s}\), and its vertical velocity falls by \(10\,\text{m/s}\) every second. The end P is a different story. It circles C three times, and whenever it passes straight above C it even moves backwards for an instant.
So the point-mass model answers one question well: where does the body as a whole go? It cannot say where the handle points, how fast it spins, or whether a leaning box will topple. For those questions a body needs size and shape. This unit builds that model and then rebuilds mechanics for it.
A spinning bat is tossed through the air. Air resistance is negligible. Which point of the bat moves along a parabola?
For which of these questions is the point-mass model enough?
The rigid body
The simplest body with size and shape is the rigid body. Picture it as a great many particles, held together so firmly that none of them can move closer to or further from any other. The body may move and turn as a whole, but it never stretches, bends or squashes.
A rigid body is a body in which the distance between any two particles stays constant, whatever forces act on the body. If A and B are any two particles, the length AB never changes.
No real body is perfectly rigid. A steel rod stretches by a tiny fraction of a millimetre when it is pulled hard. A cricket ball squashes for an instant when it hits the bat. The model is useful when these changes of shape are far smaller than the motions we study. A door, a wheel, a ladder and a spinning top all qualify. A rubber band, a spring and a wave on a rope do not; they are deformable bodies.
Which of these is closest to a rigid body?
What rigidity does to velocities
Rigidity fixes distances, and that fixes something about velocities too. Take two points A and B of a rigid body. Suppose the component of B's velocity along the line AB were larger than that of A, pointing away from A. Then B would be moving away from A faster than A follows it, and the distance AB would grow. A rigid body does not allow that.
The same argument works in the other direction. So the components of the two velocities along the line joining the points must be equal. The components perpendicular to the line are free. They may differ in size, and that difference is exactly what turns the line AB.
For any two points A and B of a rigid body, the components of \(\vec v_A\) and \(\vec v_B\) along the line AB are equal. So the velocity of B relative to A, \(\vec v_B - \vec v_A\), is perpendicular to AB.
The rod in Fig 1.2 shows the rule at work. When A is \(1.5\,\text{m}\) from the wall, B is \(2\,\text{m}\) up it, since \(1.5^2 + 2^2 = 2.5^2\). The rod then makes an angle \(\theta\) with the floor where \(\cos\theta = 0.6\) and \(\sin\theta = 0.8\). Along the rod, A contributes \(0.5 \times 0.6 = 0.3\,\text{m/s}\). B must match it, so \(v_B \times 0.8 = 0.3\), which gives \(v_B = 0.375\,\text{m/s}\).
Notice what this saved. We did not need to know how the rod moves as a whole. One line of geometry and one rule linked the two ends. The same rule will return many times: for a ladder, for a rod sliding on a curve, and for a rolling wheel.
A ladder \(5\,\text{m}\) long rests against a smooth wall of a house in Pune. Its foot is \(3\,\text{m}\) from the wall and is sliding away from it at \(0.8\,\text{m/s}\). How fast is the top sliding down the wall at this instant?
The top is \(\sqrt{5^2 - 3^2} = 4\,\text{m}\) up the wall. Let \(\theta\) be the angle of the ladder with the floor, so \(\cos\theta = 3/5\) and \(\sin\theta = 4/5\).
The foot moves along the floor, at angle \(\theta\) to the ladder. The top moves down the wall, at angle \(90° - \theta\) to the ladder. Equal components give \(v_A\cos\theta = v_B\sin\theta\).
\(0.8 \times 0.6 = v_B \times 0.8\), so \(v_B = 0.6\,\text{m/s}\).
The top slides down at \(0.6\,\text{m/s}\). Check with calculus: \(x^2 + y^2 = 25\) gives \(x\dot x + y\dot y = 0\), so \(\dot y = -3 \times 0.8/4 = -0.6\,\text{m/s}\).
A rigid rod AB moves in a plane. At one instant, end A moves at \(2\,\text{m/s}\) along the rod, from A towards B. The velocity of end B makes an angle of \(60°\) with the rod. What is the speed of B, in m/s?
The ends of a rigid rod AB lie on the x axis, A at the origin and B at \(x = 1\,\text{m}\). At one instant A moves at \(3\,\text{m/s}\) along \(+x\). Which velocity of B is possible?
Explain why the velocity of one point of a rigid body relative to another point is always perpendicular to the line joining them.
The distance between the two points cannot change. A relative velocity with a component along the line joining them would change that distance at once. So the relative velocity has no component along the line, and it must be perpendicular to it. Seen from one point, the other can only move on a circle round it.
Stand at one point of a rigid body and watch another. The distance to it is fixed, so all you can ever see it do is move on a circle round you. That is why rotation appears as soon as a body has size: relative to each other, its points can only turn.
In 1765 the Swiss mathematician Leonhard Euler, then working in Berlin, published a long book on the motion of solid bodies. Newton had written laws for particles. Euler asked how a body of fixed shape moves, and he took the unchanging distances between its particles as the starting point. From that one assumption he built the equations of spinning tops and tumbling bodies. His book gave the rigid body its place in mechanics, and it also named the quantity at the centre of this chapter.
Now return to the bat tossed to the dressing room. The team-mate judges the flight of one point, the centre of mass, which moves on a parabola like a ball. Every other point of the bat loops round that centre. To describe those loops we need a body whose distances stay fixed, and the next lesson sorts its motions into three kinds.
- Treating every point of a thrown, spinning body as moving on a parabola.Only the centre of mass moves like a projectile. Other points add a turning about it.
- Equating the whole velocities of two points of a rigid body.Only the components along the line joining the points must be equal. The perpendicular components may differ.
- Using the angle with the wall where the angle with the rod is needed.Resolve each velocity along the rod itself. Draw the rod, then measure each velocity's angle from it.